Abel–Jacobi map

Remarks

The map is well defined: two paths from $P_0$ to $P$ differ by a closed loop, which changes the vector of integrals by an element of $\Lambda$. On divisors of degree $0$ the result does not depend on $P_0$.

Example: let $X = \C / \Lambda'$ be a complex torus with $g = 1$ and $\omega_1 = dz$. The integral from $P_0$ to $P$ is $P - P_0$ modulo the periods of $dz$, which form $\Lambda'$. So $A(P) = P - P_0$ is an isomorphism from $X$ onto $J(X) = \C / \Lambda'$.

Non-example: the map $P \mapsto \left( \int_{P_0}^{P} \omega_i \right)_i$ into $\C^g$ itself, without the quotient by $\Lambda$, is not well defined on $X$. Going once around a loop with a nonzero period changes its value.

Abel's theorem: a divisor $D$ of degree $0$ is the divisor of a meromorphic function if and only if $A(D) = 0$. Jacobi inversion: the induced map from effective divisors of degree $g$ to $J(X)$ is surjective. For $g \ge 1$ the map $A \colon X \to J(X)$ is injective, and for $g \ge 2$ it is a holomorphic embedding.